On the feedback vertex set polytope of a series-parallel graph

Authors:
Samuel Fiorini;Odile Marcotte
Affiliations:
Département de mathématiques, Université Libre de Bruxelles, CP 216, Boulevard du Triomphe, B-1050 Bruxelles, Belgique;Université du Québec í Montréal and GERAD, HEC Montréal, 3000, chemin de la Côte-Sainte-Catherine, Montréal, Canada H3T 2A7
Venue:
Discrete Optimization
Year:
2009

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Abstract

The minimum weight feedback vertex set problem (FVS) on series-parallel graphs can be solved in O(n) time by dynamic programming. This solution, however, does not provide a ''nice'' certificate of optimality. We prove a min-max relation for FVS on series-parallel graphs with no induced subdivision of K"2","3 (a class of graphs containing the outerplanar graphs), thereby establishing the existence of nice certificates for these graphs. Our proof relies on the description of a complete set of inequalities defining the feedback vertex set polytope of a series-parallel graph with no induced subdivision of K"2","3. We also prove that many of the inequalities described are facets of this polytope.